What is the smallest two-digit integer $n$ such that switching its digits and then adding 3 results in $2n$?
Solution: Let $10a+b$ represent $n$, where $a$ and $b$ are the tens and units digits, respectively. Switching the digits and adding 3 results in $10b+a+3$, which we set equal to $2n$. \begin{align*}
2(10a+b)&=10b+a+3\quad\Rightarrow\\
20a+2b&=10b+a+3\quad\Rightarrow\\
19a&=8b+3
\end{align*}For the smallest $n$, we let the tens digit $a=1$. We have $19=8b+3$, which means $b=2$. So the smallest $n$ is $\boxed{12}$.